15 WEAK TOPOLOGIES 77
(12.15.9) Let E be a separable normed space. Then every closed ball B':
\\x'\\ ^ r in E' is metrizable and compact with respect to the weak topology.
By (12.15.7) and (12.15.7.1) it is enough to show that B' is weakly closed,
and this follows from the fact that the function JC'H»||JC'|| is lower semi-
continuous on E' (12.7.2).
Now let E be a Hilbert space (6.2). For each xe E, let/*) denote the
continuous linear form y*-+(y\x) on E. It follows from (6.3.2) that x\-*j(x) is
a semilinear isometry (i.e., we have
for all scalars X) of E onto its dual E'. We can therefore "transport" to E the
weak topology on E' : the weak topology on E is therefore the topology defined
by the seminorms x\~+ \(a \ x)\ as a runs through E. This topology is coarser
than the topology defined by the norm on E (which is called the strong
topology of the Hilbert space E). From (12.15.9) we have:
(12.15.10) In a separable Hilbert space, every closed ball is metrizable and
compact with respect to the weak topology.
If w is a continuous endomorphism (with respect to the strong topology) of
a Hilbert space E, then by virtue of (12.15.4) and (11.5.1)
(u(x)\y) = <«(
for all x,y in E, and therefore (6.3.2)
(12.15.11) w*=7~1 o'Wo/
This relation shows immediately that '# is strongly continuous and that
H't/ll = ||w*|| = \\u\\. Replacing u by u* in (12.15.11), we see that every strongly
continuous endomorphism of E is also weakly continuous (cf, (12.16.7)).
(1 2.1 5.1 2) In a Hilbert space E, a sequence (xn) converges strongly to a point a
if and only if it converges weakly to a and Urn \\xn\\ = \\a\\.
n~+ oo
The conditions are clearly necessary. To see that they are sufficient, con-
sider the formula
Since by hypothesis the sequence ((xn \ a)) converges to (a \ a) = || a\\2, it follows
that llx. - all tends to 0.eak topology on E'.